Method of damping devices having oscillatory motion

ABSTRACT

A method for damping an oscillating mechanical system to bring it to rest at or near its equilibrium position employing iteratively a fixed damping cycle consisting of a first interval of undamped motion followed by a second interval of supercritically damped motion, the process being iterated a sufficient number of cycles to position the system as close as desired to the equilibrium position. A preferred application of this method of damping is in connection with a meridian-seeking gyroscope, where the first interval of undamped motion consists of free precession of the gyroscope pin axle toward the meridional plane, followed by a second interval of supercritical damping that positions the spin axle closer to the meridional plane than at the release of the motion. By repeating the appropriately timed intervals of the cycle a number of times, the spin axle of the meridian-seeking gyroscope can be brought as close to the meridional plane as desired.

is t 1 1 United States ate t 1 11 3,577,

[ 72] Inventor Harry Nils Eklund FOREIGN PATENTS pahsadescahf- 314,70310/1919 Germany 33/226 [21] Appl. No. 687,049 [22] Filed Nov, 30, 1967Primary Exam ner-Robert B. Hull [45] P t d M 4, 1971 Attorney-Christie,Parker & Hale [73] Assignee I.earSiegler,lnc.

1 Santa ABSTRACT: A method for damping an oscillating mechanical systemto bring it to rest at or near its equilibrium position employingiteratively a fixed damping cycle consisting of a first interval ofundamped motion followed by a second interval of [54] METHOD OF DAMPINGDEVICES HAVING supercritically damped motion, the process being iterateda OSCILLATORY MOTION sufiiclent number of cycles to position the systemas close as desired to the equilibrium position. 10 Claims, 1 1 DrawingFig? A preferred application of this method of damping is in con- U.S.nection a meridian-seel ing gyroscope where the first in. r 74/ tervalof undamped motion consists of free precession of the [5 l Int.gyroscope pin axle toward the meridional plane followed a of econdinterval of uper-critical that positions the 56 R f ed spin axle closerto the meridional plane than at the release of 1 e erences It themotion. By repeating the appropriately timed intervals of UNITED STATESPATENTS I the cycle a number of times, the spin axle of the meridian-2,802,279 8/1957 Agins 33/226 seeking gyroscope can be brought as closeto the meridional 3,125,885 3/1964 Malone 73/517 plane as desired.

JMFZM 1 Z 4 I I 7 5 5 iPO fl/z'z 0/; AMPZ/F/[C Mme i l d 1 75 (if/V. 1 Z7 1 1" ZZZ; fimmrz [my I L I I a 1 l0 /7 l I" 'l l I i i M maria-E I, AMFEE I M I mum/517ml I ('l/FfA/T l4? E 04/: mjmme Jim/name M 1 a t fi Tmam/51m I (urea/7 E l6f1/E647fl l /5 Q l i [Ar/mo: 1 #18452: I Fania/MmeTIME/C l ,0 2/ J 1 2i METHOD OF DAMPING DEVICES HAVING OSCILLATORYMOTION BACKGROUND OF THE INVENTION 1. Field of the Invention Thisinvention relates to a method and means of controlling a device havingoscillatory motion, and more particularly to damping the device to bringthe moving element to rest in, or near, its equilibrium position in arelatively short period of time, and in a more precise and dependablemanner than heretofore, which invention has particular applicability tomeridian-seeking gyroscopes.

Many devices, such as ordinary gravity pendulums, bifilar and trifilargravity pendulums, torsion pendulums, spring-centered shafts, andspring-suspended masses, have elements that oscillate or vibrate aboutan equilibrium position. Oftentimes the oscillation is undesirable, suchas for example when the device is a meridian-seeking gyroscope fordirect indication of a meridional plane. A pendulous meridian-seekinggyroscope upon the earthss surface having its spin axis horizontal tendsto precem under eanhss rotation so as to align its spin axis withastronomical north, provided the gyroscope is given angular freedomabout its vertical or azimuth axis.

Meridian-seeking gyroscopes are useful as instruments for surveying and,in general, accurate direction finding. However, their usefulness islimited because of the inability to easily and quickly bring the spinaxis in alignment with the meridian or astronomical north. In theabsence of added damping, it oftentimes takes as much as two days orlonger for the spin axis to come to rest at its equilibrium position inthe meridian plane after oscillating with respect to this plane.

2. Description of the Prior Art In the past oscillatory devices havebeen brought to rest at or near their equilibrium position by applyingsome type of damping, such as for example, viscous damping oreddy-current damping. In some of the meridian-seeking pendulousgyroscopes a continuous velocity-proportional damping has been appliedwhere the damping torque is less than the critical damping torque of thesystem. The application of this damping torque will bring theoscillatory element to its equilibrium position after a certain periodof time. However, it has been found that in many cases this period oftime is excessive and limits the usefulness of the device.

Presently used damping arrangements for meridian-seeking gyroscopesprovide a damping torque in azimuth proportional to either theinstantaneous angle of elevation of the spin axle, or to theinstantaneous angular velocity of the spin axle in the azimuth plane. Ineither case, the magnitude of the damping torque is chosen, so as toprovide somewhat less than critical damping torque, the actual magnitudebeing a compromise between that required for adequate stability of thegyroscope, after coming to rest in the meridian plane, and that requiredto bring the spin axle within the allowable error angle, with regard tothe meridian plane, in a given time interval after uncaging of thegyroscope.

Generally, such damping arrangements employ a damping ratio less thanunity (usually between 0.7 and 0.9), so that the spin axle executes adamped oscillatory motion about the meridian or equilibrium positionbefore settling down within the allowable azimuth error angle. Thesettling time for a meridian-seeking gyroscope with such a dampingarrangement may be of the order of 2 to 2 2 times the length of theundamped period of the gyroscope, depending on the actual damping ratioand the magnitude of the initial azimuth angle. However, due tovariations in the value of the damping ratio with temperature and otherfactors, this method introduces a certain element of uncertainty, as towhen the amplitude of the oscillations of the spin axle about themeridian have decayed below the magnitude of the allowable azimuth errorangle.

Thus, in accordance with this invention, devices that have oscillatorymotion about an equilibrium position are brought to rest in a relativelyshort interval of time employing iteratively a fixed damping cycleconsisting of a first interval of undamped motion followed by a secondinterval of supercritically dampened motion, the process being iterateda sufiicient number of cycles to position the system as close as desiredto the equilibrium position.

A preferred application of this method of damping is in connection witha meridian-seeking gyroscope, where the first interval of undampedmotion consists of free precession of the gyroscope spin axle toward themeridional plane, followed by a second interval of supercritical dampingthat positions the spin axle closer to the meridional plane than at atthe release of the motion. By repeating the appropriately timedintervals of the cycle a number of times, the spin axle of themeridianseeking gyroscope can be brought as closed to the meridionalplane as desired.

The method of the present invention has general applicability to anyoscillatory mechanical system with plural degrees of freedom and hasspecific applicability to systems which obey the differential equation dx da: m -l- C d -lkw 0 Such systems, which include nonpendulous andpendulous meridian-seeking gyroscopes can be represented graphically ina phase-plane diagram by writing the equation for the system in the formand is the angular natural frequency of the system, t is the time, andplotting v versus x. Phase-plane diagrams are very useful in analyzingoscillatory of vibratory systems. An example of phase-plane diagramanalysis is set out at pages 353- -363 in the book entitled Mechanicalvibrations, Fourth Edition, by JP. Den I-lartog, published byMcGraw-I-Iill Book Company.

When 8=0 the trajectories in the x, v-plane are circles, the radii ofwhich are the initial values, x of the displacement. Then 6 is differentfrom zero, and much greater than unity, the phase-plane trajectoriesbecome virtually straight lines everywhere, except in the neighborhoodof an asymptote whose equation is given by x'+(6 /5 T)x=0, which allother trajectories but one approach asymptotically and proceed along tothe stable, singular point at the origin of the phase plane. A pointP(x,v), the representative point of the state of the system, progressesin time along a trajectory defined by the initial condition of thedamped motion, and, upon approaching the asymptote, will proceed alongthis to the origin, which it will reach only after infinite time. Oneexception to this approach to the origin is the trajectory whoseequation is x +(6+ V8 1) i=0, and which represents a straight linethrough the origin. No trajectory can cross this straight line, but canonly approach it along the above asymptote and will be referred to herein a special sense as a second asymptote of the system, along which themotion rapidly proceeds tothe proximity of the singular point at theorigin. These two asymptotes are trajectories of slowest and fastestapproach respectively, to the singular point of the origin.

The method of damping the system, whose equilibrium position coincideswith the singular point in the phase plane, consists in releasing thesystem with zero velocity from its displaced position, allowing therepresentative point to proceed along the circular trajectory, definedby the initial displacement, until the representative point arrives atthe intersection between the undamped trajectory and the dampedtrajectory represented by the fast asymptote. At this point the dampingis switched on, and the representative point, P(x,v), moves rapidlyalong the fast asymptote into the proximity of the stable singular pointat the origin, which is the equilibrium position of the system.

The fast asymptote is, thus, the equivalent of a switching line, exceptthat the switching action is not initiated by the fast asymptote, butrather by determining the interval of the undamped motion so that at thetime of switching the representative point will be at the fastasymptote.

However, due to errors in the timer, it may happen that the switchingfrom the undamped to the damped state of the system occurs either tooearly or too late so that, instead of proceeding directly to the originalong the fast asymptote, the representative point will proceed alongone of the neighboring trajectories, become captured by the slowasymptote and, thus, never reach the origin, or the equilibriumposition. To remedy this failing, the original cycle, consisting of afirst time interval of free, undamped motion followed by second timedinterval of damped motion, is repeated by starting a new interval ofundamped motion from the final position of the representative point atthe slow asymptote. After completing the second cycle, the point willagain virtually come to rest on the slow asymptote. The second encounterof the representative point with the slow asymptote represents aposition closer to the origin, than at the first encounter after thefirst damping cycle. Thus,-by iterating the process a number of cycles,it is possible to get as close to the origin as desired, despite errorsin timing of the free, undamped interval of motion.

It is apparent that the slow and the fast asymptotes, which arecharacteristic of the system, serve as reference or guidelines, and thatin particular the slow asymptote will establish a fixed reference line,from which the undamped mo tion will start each cycle, thus preventing acumulative error in position of the system state point, orrepresentative point P, for a single, constant error in the timing ofthe undamped interval of motion.

This type of damping process is of particular value in the damping of ameridian-seeking gyroscope so that it will come to rest in or near tothe meridional plane rapidly and from any initial displacement.

The use of the fast asymptote and the slow asymptote as switching linesis a special case of the more general approach of employing any twoselected switching lines, one to determine the time for applying thesupercritical damping and the other to determine the time for removingthe damping and allowing undamped or proper motion.

The determination of the interval of time for the representative point Pto intercept the selected switching line at the end of the initialundamped motion is simplified if the moving element of the system startswith no initial velocity. This is accomplished by applying supercriticaldamping simultaneously with the release of the system from its initialdisplacement. This brings the slow asymptote into existence, and as aresult the representative point, which is now moving on a dampedtrajectory, cannot cross the slow asymptote, but comes virtually to restthereon, so that the undamped motion will be initiated from the slowasymptote.

The apparatus for selectively applying the damping force includes ameans for sensing the speed at which the element moves toward itsequilibrium position. The apparatus further includes a means responsiveto the sensing means and thus to the instantaneous velocity of theelement for generating a damping force that is proportional to thevelocity of the moving element. This damping force is applied to themoving element through a timer which sets the interval of time duringwhich the damping force is applied and also sets the interval of timebetween the applications of the damping force, when iterative damping isemployed.

The time interval between the applications of the damping force, as wellas the magnitude of the damping force are adprimarily proportional tothe instantaneous velocity below a first selected magnitude of velocityand primarily proportional to the square of the instantaneous velocityabove a second selected magnitude of velocity.

BRIEF DESCRIPTION OF THE DRAWINGS The above and other features andadvantages of the present invention may be understood more fully andclearly upon con sideration of the following specification and theaccompanying drawings in which:

FIG. 1 is a phase-plane representation of the normalized motion of anundamped oscillatory or vibratory element;

FIG. 2 is a phase-plane representation of the normalized motion of anoveHlamped or supercritically damped moving element having oscillatorymotion;

FIG. 3 is a block diagram of the damping apparatus in accordance withthe present invention;

FIGS. 4 and 5 are graphs of the damping ratio versus time of theiterative damping in accordance with the present invention;

FIG. 6 is a phase-plane representation of the general method ofiterative damping in accordance with the present invention;

FIGS. 7 and 8 are phase plots graphically depicting a preferred methodof iterative damping in accordance with the present invention;

FIG. 9 is a phase-plane representation of the motion of an undampedelement and a supercritically damped element along the fast asymptote;

FIG. 10 is a phase-plane representation graphically depicting theiterative damping in accordance with the present invention for a movingelement having a negative initial angle; and

FIG. 11 is a phase-plane representation of large initial angle undampedmotion of a moving element which has a restoring force that isproportional to the sine function for the displacement.

DESCRIPTION OF THE PREFERRED EMBODIMENTS Devices that have elements thatoscillate or vibrate about an equilibrium position may be analyzed moreeasily by considering mathematical equations for the motion of theelement and graphical plots of the solutions of the equations for themotions of the device.

In one embodiment of this invention the device having an oscillatoryelement having periodic motion about an equilibrium position is ameridian-seeking gyroscope, although the invention is in no way limitedto this particular device for it is equally applicable to any devicehaving oscillatory motion, such as ordinary gravity pendulums, bifilarand trifilar pendulums, torsion pendulums, spring-centered shafts andspringsuspended masses, for example. However, a meridian-seekinggyroscope will be employed in describing the invention.

In general there are two types of meridian-seeking gyroscopes, either ofwhich could be used in describing the invention. These are the pendulousgyroscopes and the nonpendulous gyroscopes. The pendulousmeridian-seeking gyroscope will be employed in this description.

Such a gyroscope includes a spinning rotor, spin axle, suitably mountedin a stator frame, with means for keeping it spinning. The gyroscopeproper is enclosed in frame, which is a pendulous horizontal axisgyrocompass has 2 of freedom, 1 of freedom about an elevation axis andanother degree of freedom about an azimuth axis. In using a pendulousgyroscope to describe the invention, the spin axle movement will betaken to be synonymous to movement of the frame enclosing the gyroscopeproper of the illustrative gyroscope.

For small angles of displacement the equations of motion of a dampedpendulous meridian-seeking gyroscope may be written in the simplifiedform a da and (N where C is the damping moment coefficient about theazimuth axis; ml; is the azimuth angle, with the angle being positive tothe east; His the angular momentum of the gyro rotor about the spinaxis; 0. is the local horizontal component of the earthss spin velocity,w 9 is the elevation angle, which is positive upward; M is the pendulousmoment about the elevation axis; and t is time.

In these equations, 1 and 2, the damping about the elevation axis andthe moments of inertia of the gyro container and the gyro wheel aboutthe two axes perpendicular to the spin axis have been neglected. Thesequantities are negligibly small compared to the remaining quantities,and are generally ignored in analyzing the performance of a pendulousmeridian-seeking gyroscope. The absence of these quantities in thefollowing mathematical equations in no way invalidates the results ofthe analysis, but makes the system amenable to phase-planerepresentations. Additionally, it is convenient to normalize equations 1and 2 by introducing nondimensional time defined by Y= n t, where w, isthe natural undamped frequency of the system given by 2 M2 H and bysetting 26 H (2 and 6 is commonly called the damping ratioand is definedas the ratio of the actual damping moment coefficient, C, to thehypothetical damping coefficient having the value 2H/w,,,

which hypothetical damping coefficientproduces critical damping. Whenthese substitutions are made in equations 1 and 2 the following results:

and

As is well known, the end of a spin axle of an undamped gyroscope tracesan elliptical path in a vertical plane which is perpendicular to themeridional plane with the major axis in the azimuth plane and the minoraxis in the meridional plane. With the substitution F dr 0 (5) Equations5, which represents the azimuthal motion, is a second order differentialequation having a single parameter 6, which denotes the damping ratio ofthe system. Thus, 8 l represents an under-damped system, the solution ofwhich can be expressed by a damped sinusoid. 6=l results in criticaldamping which gives rise to a damped linear algebraic function, while 81 represents an over-damped or supercritically damped system. Finally,in the case where 8- 0, the solution is an undamped sinusoidrepresenting, for small angles, a simple harmonic motion.

The solution of the equations, 5 and 6, for the undamped case and forthe supercritically damped case, where 6 l, are respectively given byfor 5 1, where 111,, is theiritial azimuth angle g istheiniial elevationangle, and 1 t 6 -1.

Equation 5 is a general equation for a damped linear oscillatory systemthat has a single degree of freedom. A singledegree-of-freedom springsystem is an example of such a system. A discussion of this system andthe resultant differential equation is set out in the Den Hartog bookMechanical Vibrations at pages 24 to 26. Additionally, equation 5 is thesame for both pendulous and nonpendulous meridan-seeking gyroscopes fortheir azimuth motion, where n11 is the azimuth angle and 8 is thedamping ratio about the azimuth axis. For a nonpendulous gyroscope,however, the natural frequency of the undamped motion is given by whereI is the moment of inertia about the azimuth axis.

For purposes of better illustrating the present invention the behaviorof the gyroscope is depicted in a special type coordinate plane known asthe phase plane in which the nondimensional azimuth velocity, dill/d7,of the spin axle is plotted against the azimuth displacement, ill, ofthe spin axle. Phaseplane diagrams are useful in analyzing oscillatorysystems and a specific example is set out at pages 353-363 of theabovereferred to book by Den Hartog. By introducing the abbreviation l II d'r equation 5 can be rewritten asfollows:

which has the well-known solution A ip -hlFaxC Wh constantotl th t i i ii..- determined by the initial conditions. Thus, if at r=0, \l1=0 and1l1=rl1 this constant, C, will be equal to and equation 9 becomes lIJ'i'IIJ B-X JI (9a) This is the equation of a family of concentric circlesof radius 41,, and corresponds to the undamped precessional motion ofthe gyro due to the action of the pendulous moment in conjunction withthe earthss rotation.

The phase-plane representation of equation 9a is depicted in FIG. 1 withcurves a, b, and c representing some of the possible solutions. In FIG.1, the representative point P moves clockwise with constant angularvelocity along the circular trajectory of radius 41 The instantaneousposition of point P is defined by the angle 1', which, according toconvention, is negative. The instantaneous values of i1; and 104 arethus given by rlFrll cos -r and tla'qp sin 1'.

An alternative way of considering the undarnped precessional motion ofthe spin axis of a meridian-seeking gyroscope would be to plotdisplacement {which is proportional to the displacement along theelevation axis, against the displacement 111 along the azimuth axis,that is, to plot the behavior of the gyro using the r11, 5 coordinateplane. Since g equals negative rlr' (equation 6), this coordinate planemay be represented by taking FIG. 1 and rotating the phase-planerepresentation through 180 about the 111 axis. This will produce themore familiar :11, 5 plane, in which the motion proceedscounterclockwise, and the angle 1 is positive.

For supercritically damped systems, where 6 l it will be seen that thesolution for equation 7 results in a phase-plane representation that isdrastically different from the plot where 8=0, which is depicted inFIG. 1. The solution curves of equation 7 for 8 1, form a family oftrajectories as depicted in FIG. 2. These trajectories are virtuallystraight parallel lines, such as lines dh in FIG. 2, except in theregion surrounding the straight line marked Al-Al. The equation for lineAl-Al is rl1+ (6V8 l)=0. This line as an asymptote that cannot becrossed by any solution curve and all of the latter will thereforeapproach this line asymptotically and will then move along it toward thesingular point at the origin, 20.

For example, as shown in FIG. 2 a trajectory point P starting at giveninitial conditions rial/ dial/ will travel on the trajectory f passingthrough the point r11 r11 approaching Al-Al asymptotically while movingtoward the origin 20, which is the point of stable equilibrium.

The motion along asymptote Al-Al is the slowest approach to the origin,which approach is representative of the creeping motion generallyencountered in overdanrped systems.

In the present invention a supercritical damping ratio is employed. Forillustrative purposes it is assumed that this damping ratio is 10 timesthe critical damping. Such a large value of damping ratio as 8 10,causes the motion along the line Al-Al to be so slow as to make thesettling time of the gyro hundreds of times longer than can bepractically tolerated. However, as will be described hereinafter, theotherwise undesirable characteristic of the slow asymptote is employedin the method and apparatus of the present invention to aid in bringingthe oscillatory or vibratory element to rest at or near the equilibriumpoint in a relatively short interval of time.

The trajectory along the line A2 A2, which passes through the origin 20,as shown in FIG. 2, constitutes a trajectory of fastest approach to theorigin. Thus, if a trajectory originates on this asymptote, A2A2, or isby some means made to enter thereupon, the representative point P willmove rapidly along the asymptote toward the origin and reach theproximity thereof in the shortest possible time. The equation for lineA2-A2 is Since the origin is a stable singular point, the representativepoint P travelling along the fast asymptote comes to rest in theproximity of the meridional plane, which plane passes through the originand is perpendicular to the :11 axis.

The slow and fast asymptotes Al-Al and A2-A2, respectively, in FIG. 2are displaced from the axes Ill, 104' by an angle a. This angle a isdirectly dependent upon the selected damping ratio 8, and since bydefinition,

1 Cam 8 is directly dependent on the basic characteristics of thesystem. As the damping ratio 5 increases, the angle or decreases and foran infinite damping ratio 8, a would be zero so that the fast asymptotewould coincide with the ill axis and the slow asymptote would coincidewith the 111 axis of the phase-plane representation in FIG. 2.

As depicted in the phase plane, where it: is assumed to be. 10, FIG. 2,the family of curves resulting from the solution of equation 7 consistsof trajectories that are virtually straight parallel lines, except inthe vicinity of the slow asymptote, Al-Al. The trajectories for 8 10 aremore nearly perpendicular to the ill axis than shown in FIG. 2 and thesubsequent FIGS. of the drawing. This is because the angle a isexaggerated to show more clearly the relationship of the terms of theequations. For example, the angle a in FIG. 2 is depicted as beingbetween 6 and 7, while in actual practice the angle is less than 3 for adamping ratio 8 10.

As the damping ratio 8 is increased above the assumed value of 10, thetrajectories become straighter and more parallel to each other up to thepoint where the damping ratio is infinite and the trajectories becomeparallel to the :1! axis. Conversely, as the damping ratio is decreasedbelow 10, the family of trajectories become less straight and morecurved up to the point where 8=0, with a resultant family of concentriccircles. Trajectories for 8 are shown in FIG. 8.23 on page 356 of theabove-identified book. The shortest possible time in which the gyro spinaxis can be brought to the meridian plane, as I seen by combining FIGS.1 and 2, is to bring the representative point P along a circulartrajectory representing undampened motion onto the fast asymptote A2-A2and then allow point P to follow this asymptote to the origin, which islocated in the meridian plane.

An apparatus for performing the method of the present invention isdepicted schematically in the block diagram of FIG. 3. For illustrativepurposes, the device to be controlled is assumed to be ameridian-seeking gyroscope having precessional motion with displacementabout both azimuth and elevation axes.

The operation of the control system depicted in FIG. 3 is described withreference to the phase plane as representatively depicted in FIGS. 1 and2. The azimuthal movement of the spin axle of the gyroscope is set forthby equation 7 and conforms to the solutions of this equation, when 8=0and EFIO as depicted in FIGS. 1 and 2, respectively. The inputs to thegyroscope, which are shown schematically on FIG. 3, are g and 2 8111 towhich the gyroscope responds with the outputs 1!; and til. When 8 equalszero the only input is g and the gyroscope executes a simple harmonicmotion, the precessional motion depicted in FIG. 1. On the other hand,for 8 equal to 10, the motion of the gyroscope, which is described bythe representative point P, will be along one of the trajectories shownin FIG. 2.

The control system operates on the azimuth motion of the spin axle,which is the oscillatory of vibratory element of the meridian-seekinggyroscope 1 in FIG. 3. The control system includes a pickoff 1 coupledto the azimuth motion of the spin axle of the gyroscope 1 and has anoutput representative of the displacement and velocity of the gyroscopespin axle in the azimuth plane. The output of the pickoff 2 is amplifiedby amplifier 3 and applied to motor 4. The output of the motor 4 iscoupled back to the input of the pickoff 2 through a speedreductionmechanism, such as a gear box 5. The displacement output of the gear boxcombines with the displacement output from the gyroscope to form asumming junction 6. The coupling between the gear box 5 and the motor 4and the gear box 5 and the summing junction 6 is mechanical, whichmechanical coupling is shown by dotted lines in the drawing.

Thus, the azimuth motion of the gyroscope 1 is followed up by aservosystem consisting of pickoff 2, amplifier 3, motor 4, and gear box5.

A typical pickoff and servoloop for a meridian-seeking gyroscope isdescribed in the copending US. Pat. application Ser. No. 529,325, filedFeb. 23, 1966, by Leonard R. Ambrosini and assigned to the same assigneeas this application now U.S. Pat. No. 3,512,264. For purposes ofillustration, it will be assumed that the gyroscope l, pickoff 2,amplifier 3, motor 4, and gear box of FIG. 3 are similar to thecorresponding elements in the referenced application.

The output of the motor 4 is alsoapplied to a tachometergenerator 7, theoutput voltage of which is applied to an amplifier and demodulator 8.The output voltage of the tachometer-generator 7 is representative ofthe angular velocity and is proportional to the angular velocity of thespin axle in the azimuth plane and is applied to a timer and dampingcontroller 9 for generating a damping torque about the azimuth axis,which torque is proportional to the instantaneous velocity of the spinaxle in the azimuth plane. The damping torque is applied to theoscillatory element, that is'to the gyroscope 1 through a torquer l0.

In the servo described above, the gear box 5 allows the motor 4 to runat a relatively high speed for more uniform motion. This relatively highspeed also increases the output voltage of the tachometer-generator 7mounted on the shaft of motor 4. The relative displacement between thegyrodriven part of the pickoff 2 and the motor-driven part of thepickoff 2 generates an error signal 6. This error signal is amplified byamplifier 3 and applied to the motor 4 with such a polarity that itreduces the pickoff error signal. Phase and amplitude compensation mayalso be provided in the amplifier to improve the servoresponse.

Additionally, the stability of the servo may be improved by applying thevelocity feedback signal from the output of the tachometer-generator 7through a feedback network 12 to the input of the amplifier 3 as shownin FIG. 3.

The output voltage of the tachometer-generator 7, which is proportionalto the azimuth velocity dill/tit, is amplified and demodulated in theamplifier and demodulator 8 and then fed to the timer and dampingcontroller 9, which controls the inputs to the gyroscope torquer 10. w

The gyroscope torquer 10 is depicted to FIG. 3 by coils 13 and 14, whoseaxes are at right angles to one another. Coil 13 is mounted on thegyroscope housing and carries a constant current I. Coil 14 is mountedon the servo followup of the gyroscope and carries a variable current Iwhich is proportional to the tachometer-generator 7 output voltage,which is proportional to the output velocity dtlI/dt of the gyroscope.Coil I3 cooperates with coil 14 so that when both coils areappropriately excited a torque will be developed about the azimuth axisof the gyroscope. The gyroscope torquer 10 thus furnishes the dampingterm (28111) about the azimuth axis of the gyroscope. The torque exertedby one coil on the other is given by T=C l I where C is a constant ofproportionality. Since I is constant and I is proportional to 111', itis evident that T=C \1:', where C is another constant ofproportionality. By giving I the proper value so that C =C,I =28, onethus achieves that T=28t,l1', which is the desired instantaneous valueof the damping torque to be impressed upon the gyroscope. While thedamping torque is shown in FIG. 3 as being applied to the gyroscope ormoving element by way of a torquer, it could be applied in any suitablemanner. For example, it could be applied by use of some type of coulombdamping or viscous damping.

After the gyroscope has reached its operating speed and has generallybeen oriented in the direction of the meridian by Timer and dampingcontroller 9 includes a constant current generator 16 and a variablecurrent generator 17, which is responsive to the output voltage of thetachometer-generator 7 through the amplifier 8 and thereby responsive tothe instantaneous velocity dull/dz of the gyroscope about the azimuthaxis. The outputs of the generators 16 and 17 are applied to the torquer10 through transmission gates 18 and 19, respectively. The conductionstates of the transmission gates 18 and 19 are controlled by the outputof a variable timer 21. When the variable timer has an output signal ofa particular polarity the gates will be placed in their conduction stateto pass the signal from the associated current generator. In thismanner, the duration of the interval of each damping pulse may becontrolled as well as the interval between the damping pulsesbyprogramming the variable timer 21. Additionally, the application of thefirst damping pulse may be controlled and may be timed to take place atthe moment of uncaging, as representatively shown in FIG. 4, or at somelater selected time, as shown in FIG. 5. In the charts of FIGS. 4 and 5,the application of the damping ratio 8 is shown on a time scale, thetime intervals being predetermined by the variable timer 21. The curves22 and 23 of FIGS. 4 and 5 occur at the same time as the output signalfrom the variable timer 2l which opens the transmission gates 18 and 19.The time scale and the intervals of time on FIGS. 4 and 5 are shown foran illustrative meridian-seeking gyroscope having a natural undampedperiod of 240 seconds at a particular latitude.

Ideally, during the initial period after uncaging, the gyroscopeprecesses without damping for the length of time required for therepresentative point P on the circular trajectories of FIG. 1 tointersect the fast asymptote A2-A2 of FIG. 2. At this time, the fulldamping torque is switched on by the timer, and as a result, therepresentative point P travels along the fast asymptote A2-A2 into themeridian plane where it comes to rest.

The method includes in the preferred case the use of the asymptotes, inthe displacement-velocity representation (FIG. 2) of a supercriticallydamped system, as reference lines or switching lines, to determine whenthe damping should be applied or removed. The asymptotes, which aredefined in the displacement-velocity plane of the motion by themagnitude of the supercritical damping ratio of the system, pass throughthe origin of the displacement-velocity plane and have the slopes of(8\/6 l)' and (8+\/8 l respectively, for; the fast and the slowasymptote. For high supercriticalrnpifigi fi' 1 y fi'r f rms Is e s 9 y.sma angle with the displacement axis, while t h e f a st asymptote formsan equal angle with the velocity axis. Motion originating on or enteringupon the slow asymptote requires infinite time to reach the equilibriumposition, generally known as a creeping not ce. whi mst o al nsthe fssxwlatszts reaches the circular trajectories of FIG. 1 and the familyof trajectories for some other means, for example a magnetic compass,the

8 10 of FIG. 2. The lines Al-Al and A2A2, respectively, represent theslow and fast asymptotes, for a given supercritical damping ratio.

S1 and S2 are switching lines passing through the origin of the 111, allplane and forming an angle y between them. The angle 7 is proportionalto the time interval of undamped precessional motion along a circulartrajectory from S1 to S2 in the phase plane for small angles of i o,such that the system is essentially linear. A timing sequence inaccordance with the present invention is provided whereby the gyroscopeis uncaged at an initial azimuth angle 41,, and then allowed to precessundamped through an interval T at the end which the undamped precessioncircle 30 intersects switching line S2 at point C At this point thesupercritical damping is switched on and the representative point Pproceeds along the damped trajectory 31, which is parallel to the factasymptote, A2-A2, until at 8,, it is intercepted by the switching lineS1. The damping is now removed and the gyro allowed to precess freelywith the representative point following a circular trajectory 32 for aninterval given by the angle 7 between the switching lines S1 and S2, atthe end of which interval it is intercepted by the switching line S2 atpoint C,. With accurat e ti rni n g the time intervals corresponding tothe arcs B C B,C,; B C are all equal. Similarly, the damped trajectorysegments C 8 C 8 (YB etc. all correspond to equal time intervals. As aconsequence, the representative point will follow the zigzag pathbetween the switching lines until after a sufficient number of cycles itreaches the proximity of the meridian plane.

The ratio between the azimuth displacement amplitudes of any twoconsecutive cycles is constant. Thus,

One can then express the amplitude after n complete cycles by rim (w cosb) p where p is the angle between the switching line S1 and the illaxis. One notes from FIG. 6 that I n-I- 1 p I n p constant will decreaseas I gets larger and y gets smaller, resulting in a greater reduction inamplitude after a given number of cycles. However, unavoidable errors intiming make such an improvement illusory since the absolute error in atiming interval could under these conditions, even be equal to thelength of the interval itself. As a result, the switching lines will notremain in their fixed, predetermined positions, but will shift relativeto each other, in such a way as to cause the magnitudes of the angles Iand 'y to vary in a prohibitive and indeterminate manner. To overcomethe possibility of errors in timing, the slow asymptote is alternativelyused as one of the switching lines. In this way, every undamped intervalwill be commenced from a fixed reference line.

Although the use of the slow asymptote as a reference for the initiationof the undamped intervals tends to ensure that the end of the undampedinterval and the beginning of the following damping interval will occurat the point where the trajectory of the undamped precession intersectsthe fast asymptote, other circumstances may conspire to end the undampedprecession interval either before or after intersection with the fastasymptote. Thus, the timer will be subject to errors, for example, thelocal latitude is not always accurately known, the period of thegyroscope increases as the initial amplitude gets larger, etc., all ofwhich adds up to the fact that seldom, if ever, will the dampinginterval be initiated on the fast asymptote, when it is selected as aswitching line.

Thus, even when the fast asymptote is selected as the switching line forthe application of damping, the damping is made iterative, i.e., thecycle, made up of the undamped interval and the damping interval, isrepeated a sufficient number of times, so as to reduce the amplitude ofthe gyroscope to an acceptable value. The iterative damping, with theslow and fast asymptotes used as switching lines, is depictedgraphically in FIGS. 7, 8, and 10.

In FIG. 7, it is assumed that the error, Ar (corresponding to an errorof AT/wo in real time), in the undamped time interval is negative, sothat the damping interval is initiated prematurely, at B, an angle A'rahead of the fast asymptote, thus making the switching line S3 passthrough B. As a result, the damped motion now takes place along thetrajectory 40 which passes through point B. The magnitude of the errorangle A7 is exaggerated in FIGS. 7, 8, and 10, similar to theexaggeration of the angle a for ease of drawing. These angles are onlyillustrative and are not limiting.

The trajectory 40 is virtually parallel to the fast asymptote A2-A2, andis captured" at point C by the slow trajectory Al-Al, on which it comesto a virtual stop and then creeps toward the origin at an extremely lowrate. A few seconds after the representative point has been captured bythe slow asymptote, the damping is switched off by the timer, and thegyroscope resumes its free precession, with the representative point Pnow moving along the circular trajectory 41 or are CD. Since the angleWW3, it is clear that, at the end of the undamped time interval tlrepresentative point P will be at D on the switching line OB, or S3.This switching line S3 is defined by the equation ill-H11 tan(o:+A'r)=0, while the ideal switching line, i.e., the fast asymptote hasthe equation 1H1]; tan aa=0. At the end of the undamped time interval,the. timer 21 again switches on the damping and the representative pointnow proceeds along the damped trajectory 42 through D, until it iscaptured at E by the slow asymptote. The iteration may be continued inthis manner, until the amplitude of the gyroscope oscillation has beenreduced below the magnitude of the allowable error.

When the timing error is positive, and the undamped trajectory extendsan angle A1, beyond the fast asymptote, the damping process will havethe appearance shown in FIG. 8, which is self-explanatory in view ofFIG. 7, with the second switching line now being defined by theequation: (Ix-H11 tan (a-A =O.

In order to determine the attenuation of the gyroscope amplitude after acertain number of iterations of the damping process, one may apply thefollowing considerations: the angle a is less than 3 for 8=10, so thatby reference to FIGS. 7 and 8, it is seen that -*,,A-r or Since thisratio is constant for a given error, A1, one obtains readily theresidual amplitude of the gyroscope oscillation after n iterations:|11,,=t!1,,( Ar)".

For an exemplary meridian-seeking gyroscope one can, for the purpose ofillustration, assume that one-quarter period of undamped precession willrequire about 60 seconds. The transversal of the arc between the slowand fast asymptote then requires, approximately 56 seconds for a dampingratio of 10. Assuming the very pessimistic value of Ar=0.05 rad., orAt=seconds, the undamped interval will be about 56 seconds i 2 seconds.From both analytical and graphical investigations, it has, furthermore,been found that 12 seconds of time is adequate for the damped interval,as well as for the initial damping interval. The time interval sequenceto be generated by the timer 21 will thus have the appearance shownschematically in FIG. 4, where the negative timing error, A1, of minus0.05, or the equivalent of minus 2 seconds has been assumed.

If the initial amplitude is, say 5, the amplitude of the gyroscope afterfour iterations of the damping cycle will be (0.05)3.125 l0 degrees==0.l seconds of arc. In fact for this small initial amplitudesufficient accuracy is obtained in most cases with only three cycles ofiteration, since =(O.05) =2 seconds of arc. Furthermore, for manyapplications even two iterations, for which l1 =40 seconds of arc wouldbe adequate.

The interval for damping along the fast asymptote for the illustrativegyroscope having a natural undamped period of 240 seconds may becalculated by reference to the phase plot of FIG. 9.

The two asymptotes on the phase plot are derived in the followingmanner. The diagram depicts the solution for the equad b Setting W andsolving equation (7) for ill, one obtains which, for any constant valueof m, defines a straight line through the origin of slope l/m+28. Thus,in the 111, (111 plane (the phase-plane), every solution curve ortrajectory must cross the isocline, given by equation (12), with theslope m. lsocline is the mathematical term for a curve, such that whencrossed by a family of trajectories, every trajectory crosses the curveat the same slope with respect to the coordinate axes. By assigningdifferent values to m, a family of isoclines results, the m-values ofwhich define the slope of the trajectories that cross them.

There are several specific values of m that are of special interest.Thus, the isocline for m= is iIJ=0, i.e., the Ill-axis, and alltrajectories must cross this axis at right angles to it. Of particularinterest is the case in which the directions of the trajectories are thesame as the direction of the isocline, or

Substituting this value of lll'llll in equation (12) gives a seconddegree algebraic equation in m, the roots of which are Substitutingthese two values of m in equation 12) gives the two isoclines Theasymptotes are drawn on the phase-plot of FIG. 9, with the slowasymptote being line 45 and the fast asymptote being line 46.

Assuming a representative point P starting at the intersection of thecircular trajectory for undamped precession with the fast asymptote andthe application of damping at this time, the time interval for dampingcan be calculated.

The equation (l2bii) for the fast asymptote, line 46 can be written inthe form and solving this for r H l (14). or 1 At the beginning of thedamping interval, r= and (Fit/(0), which makes The dimensionless timealong the fast asymptote becomes then Since a. is small, less than 3 for6= 10, one rnay set 1 can be approxi- 1 1 mated by 1 so that 6-6 1 -6-6+tan a a=611 Again, because of the smallness of a, one can write fi 'po26 Consequently, the dimensionless time 7' along the fast asymptote isgiven in terms of the distance 1!: from the origin:

(17) corresponding to the real time.

0 In I t: o( +n) 18 system with an initial azimuth angle of radian. The\lJ-axis position of one second of arc is then, approximately, 5X 1 01110.

Substituting this value of III and the natural undamped period for thefrequency 000 in the expression for t, (equation- 18) the followingresult is obtained:

(10,; o. we; W 93 T (10,000 21r(6+11) a (20) For the i11ustrativ $6having a natural undamped period T of 240 seconds and a damping ratio of10, the time to reduce the displacement from 0.1 radians to 1 second ofarc is F133 seconds.

For the example of an initial displacement of \b,,=0.l radians and aresidual displacement of tl1=l second of arc the ratio 0' between thedamped interval of time and the undamped interval of time may be written20,000 T In The general expression for the ratio between the dampedinterval of time and the undamped interval of time is The above analysisalso holds true for negative azimuth angles. A phase representation ofiterative clamping for an initial negative azimuth angle and a negativetiming error is shown in FIG. 10.

All of the previous considerations and results have been based on theassumption that the initial angle, 41,, of the gyroscope oscillation issmall, say less than 5, so that the undamped precession angle, 1- isindependent of the magnitude of 110, that is, the system is assumed tobe linear. Actually, this is of course not generally true as themagnetic compass used for rough alignment of the meridian-seekinginstrument with the meridian, may have a large unknown declination atthe point of setup of the instrument, and as a result the initial angle1b,, may become too large to justify the assumption made in setting upequation 1, namely that sin tl;=\11. Consequently, the more exactequation for the undamped precessional motion of the gyroscope, in thephase-plane coordinates, is

d I I I +s1n I 0 (22) If this equation is integrated twice and solvedfor the dimensionless time, T, one obtains the undamped precessioninterval between the two asymptotes:

The above integral can be solved in terms of elliptic functions, butthis rather lengthy procedure can be evaded by resorting to anapproximation based on the following reasoning: Since the angle a issmall, of the order of 3, one may, for the sake of simplicity, insteadof considering the arc consider the full arc (1r/2), i.e., the quarterperiod. If one now compares the length of the quarter period, for thecase that 111,, approaches zero with that for say, ll1,,=45, one obtainsthe difference 1 [Tol =4a-' I' o] =o =.062 radians Al g ls-r 2.37seconds Thus, at the end of the first undamped precession interval, Tthe representative point P will be lagging by an angle Ar of about 0.062radians, in addition to the assumed timing error A1,, The ratio of theamplitudes after 1 cycle of damping will then, be approximately %=A1'+An=.05+.0612=0.112

so that the initial 45 angle is now reduced to approximately 5. As aconsequence, in the next and the following damping cycles, the error,A1,, due to the initial angle, vanishes, and only the assumed timingerror of 0.05 radians is present. Consequently, after 4 cycles ofdamping, the initial amplitude of 45 is reduced to 6300 degrees =2.27seconds of arc. It now becomes evident that even with much largerinitial angles, one can achieve a rapid attenuation of the gyroscopeamplitude of oscillation. For example, taking Il1 and the same timingerror of 0.05 rad., one obtains in a similar manner after 4 cycles ofdamping r11,=0.233 0.0552 (0.05 X90X3600=l4 sec. of arc. Obviously, byincreasing the number of damping cycles to 5 or 6, adequate alignmentwith the meridian can be obtained with initial angles approaching andwith a total time for alignment of about 8 min. (n=7). This is incontradistinction to the long time required for alignment of acontinuously damped exemplary gyroscope when the initial angleapproaches 180.

FIG. 11 is a phase-plane representation of the undamped precessionalmotion of the gyro for initial angles between 0 and +1 80. The portraitis shown only for one quadrant, since the curves in the other 3quadrants are the mirror images of the adjacent quadrants. The curvesbegin to deviate appreciably from the circular form, associated withsmall initial angles, as the latter go beyond, approximately, 5. This ismade evident by the dashed curve 50, which represents r =c onst., whichcurve intersects all of the precession curves at points of equal time, 7from the time of uncaging. Thus, with the timing of the undampedprecession interval kept constant at the value T the damping will beswitched in at the point, where the particular curve intersects thecurve 50. This will cause the representative point to proceed along adamped trajectory toward the slow asymptote 51. Whether it arrives atthe asymptote before switching to the next undamped precession intervaltakes place, depends on the magnitude of the initial angle and on thelength of time that has been allotted to the damping interval. In thediagram of FIG. 11, there is illustrated the case of the gyroscope beinguncaged at an initial angle of 45, which case has been considered in thediscussion of the numerical evaluation of the settling accuracy.

In FIG. 11, the representative point P proceeds along the undampedprecessional curve 52 for the interval of time allotted for undampedprecessional movement. The damping is then applied and therepresentative point P now proceeds along a damped trajectory 53 andapproaches the slow asymptote 51. At the end of this first cycle theazimuth angle is reduced from 45 to less than 5. The damping cycles arethen repeated until the oscillatory element is within the allowableerror with respect to the meridian plane. It should be noted that forlarge angles, neither the asymptotes nor the damped trajectories arestraight lines. However, this is of no consequence for a qualitativeevaluation of the behavior at large initial angles.

For purposes of illustration, it has been assumed that the damping ratio6=10. However, this damping ratio may be varied over a wide range.

The larger the damping ratio becomes, the shorter will be the timeinterval required to deprive the gyroscope or other oscillatory elementof the velocity gained during the undamped precession interval. Thus, inthe limit, as 6 approaches infinity, the trajectories of FIG. 2 becomesparallel straight lines, perpendicular to the Ill-axis, while the slowand fast asymptotes become coincident with the ill-axis and 1l1'-axis,respectively.

This condition can, of course, not be realized. Furthermore, it does notappreciably shorten the time of a single cycle. One notes that theundamped precession interval is now while the interval along the dampedtrajectory is zero. Thus, for the previously considered gyro period of240 seconds, the time of the undamped precession interval,

will be 60 seconds, as compared with 54 seconds for 8=l0, as-

suming a negative timing error of 2 seconds. Thus, for four iterationsafter an initial damping interval of i2 seconds, the total saving intime would be 36 seconds.

Although this case is only academic, it indicates nevertheless that verylittle time is saved by using even a very large value of 8. This is, ofcourse, due to the fact that most of the time required for a completecycle is used up in the more or less constant undamped precessioninterval. Furthermore, the power required for the damping torquesincreases at least as 8 Thus, increasing the magnitude of the dampingratio from 8=l0 to, say, 8=50, would require at least times more power,as well as larger coils, in order to effect a time saving ofapproximately 13 percent.

Consider now the case that the damping ratio be reduced from 8=l0 to,say 8=2.5. This increases the time along the damped trajectory by about4 times that for 8:10, while the length of the undamped precessioninterval reduces to about torque 1S given by T on 27) where the value ofthe constant k is appropriately chosen so as 44 seconds for A1=0. Thetime for a complete cycle including the initial damping interval is now(4) (12)+44=92 sec. This is an increase of approximately percent overthe 68 seconds per cycle for 8=l0, which increase would be unacceptablein many applications.

Thus, one can conclude from both practical and theoreticalconsiderations that, although no optimum value of the damp ing ratioexists per se, an optimum range of the damping ratio may be found toexist between, say 8=5 and 8=20, although 6 may be any value over 1. Theassumed value of 8=l0, used in describing the iterative damping process,can, therefore, be considered representative of this process.

When the meridian-seeking gyroscope is used at different latitudes, itwill be necessary to adjust the timing and damping control so as tocompensate for the change in latitude. However, such compensation neednot be exact; an error of il" in latitude can be easily tolerated. Theneed for the compensation of the tinting control is due to the fact thatthe period of the gyroscope is a function of the latitude:

H Constant Mm cos A Vcos A (24) so that in order to avoid an error inthe undamped precession interval, it is necessary to vary the latterwith latitude in the same way as T varies, i.e., instead of making theundamped precession time interval t,,, constant, one makes it vary inthe same manner as T:

For this purpose, a latitude compensator 25 is provided in the controlsystem of FIG. 3 which will allow the same proportional adjustment ofeach of the individual undamped precession time intervals depicted inFIGS. 4 and 5. This can be achieved in several different ways, dependingon the type of timing device employed. With mechanically orelectromechanically driven timers, the speed of the timer may be variedin inverse proportion to the gyroscope period, i.e. in proportion toVcos X, so as to expand or contract the timing sequence in acorresponding manner. In the case of an electronic timing device, eithera variation in the electronic time constants, or a variation in theintegrator voltage of a timing device, proportional to the gyroscopeperiod, may be utilized for the adjustment of the timing control.

Although the damping control system and method have been described onthe basis of employing torque linearly proportional to the velocity ofthe azimuth motion, damping torques that are not linearly proportionalto the azimuth velocity may also be used. One may, for example use adamping torque proportional to the square of the azimuth velocity a= l ll (26) to suit the requirements.

Various changes may be made in the details of construction of the systemand in the method without departing from the spirit and scope of theinvention as defined by the appended claims.

I claim:

1. A method of damping the motion of a mechanical system element whichis oscillating about an equilibrium position, the motion of the elementhaving a substantially undamped natural period of oscillation, themethod comprising the step of applying supercritical damping force tothe element for a selected interval of time commencing from when theratio of the instantaneous velocity of the element toward itsequilibrium position to the instantaneous displacement of the elementfrom its equilibrium position is approximately equal to the ratio of 1rto the product of the undamped natural period of oscillation and thedamping ratio.

2. A method in accordance with claim 1 including the additional steps ofrepeating the step a selected number of times so as to further damp themotion of the element.

3. The method in accordance with claim 1 wherein the selected intervalof time, t, is given by the expression Po 1"[111 -111 tp] 2145 5 1undamped period, ill, is the initial displacement from the equilibriumposition, '11 is the displacement from the equilibrium position at theend of said selected interval of time, and 8 is the supercriticaldamping ratio.

4. The method in accordance with claim 1 wherein the supercriticaldamping force is proportional to the velocity of the moving element.

5. The method in accordance with claim 1 wherein the supercriticaldamping force is proportional to the square of the velocity of themoving element.

6. The method in accordance with claim 1 wherein the supercriticaldamping force is a function of the velocity of the moving element.

7. A method in accordance with claim 1 wherein the supercritical dampingratio is approximately 10.

8. A method in accordance with claim 1 wherein the supercritical dampingratio is between 5 and 20.

9. The method of damping in accordance with claim 1 including inaddition a first initial step of applying supercritical damping force tothe element for a selected interval of time and a second initial step ofremoving the supercritical damping force thereby allowing the element tomove undamped until supercritical damping force is reapplied inaccordance with claim 1.

10. The method in accordance with claim 9 wherein the time at which thesupercritical damping force is removed precedes the time at whichsupercritical damping force is reapplied by an amount which isapproximately equal to the quantity t, where t is given by theexpression where 6 is the supercriticalddarnping ratio and T is theundamped natural period.

where T is the natural

1. A method of damping the motion of a mechanical system element whichis oscillating about an equilibrium position, the motion of the elementhaving a substantially undamped natural period of oscillation, themethod comprising the step of applying supercritical damping force tothe element for a selected interval of time commencing from when theratio of the instantaneous velocity of the element toward itsequilibrium position to the instantaneous displacement of the elementfrom its equilibrium position is approximately equal to the ratio of pito the product of the undamped natural period of oscillation and thedamping ratio.
 2. A method in accordance with claim 1 including theadditional steps of repeating the step a selected number of times so asto further damp the motion of the element.
 3. The method in accordancewith claim 1 wherein the selected interval of time, t, is given by theexpression undamped period, o is the initial displacement from theequilibrium position, psi is the displacement from the equilibriumposition at the end of said selected interval of time, and delta is thesupercritical damping ratio.
 4. The method in accordance with claim 1wherein the supercritical damping force is proportional to the velocityof the moving element.
 5. The method in accordance with claim 1 whereinthe supercritical damping force is proportional to the square of thevelocity of the moving element.
 6. The method in accordance with claim 1wherein the supercritical damping force is a function of the velocity ofthe moving element.
 7. A method in accordance with claim 1 wherein thesupercritical damping ratio is approximately
 10. 8. A method inaccordance with claim 1 wherein the supercritical damping ratio isbetween 5 and
 20. 9. The method of damping in accordance with claim 1including in addition a first inItial step of applying supercriticaldamping force to the element for a selected interval of time and asecond initial step of removing the supercritical damping force therebyallowing the element to move undamped until supercritical damping forceis reapplied in accordance with claim
 1. 10. The method in accordancewith claim 9 wherein the time at which the supercritical damping forceis removed precedes the time at which supercritical damping force isreapplied by an amount which is approximately equal to the quantity t,where t is given by the expression where delta is the supercriticaldamping ratio and T is the undamped natural period.